Observing the Pancharatnam-Berry phase on the
Heather Hill, Duke University,
Marty Cohen and John Noé
Laser Teaching Center,
Stony Brook University
In 1984, Michael Berry discovered a geometric phase in quantum systems
, which soon led to a whole new understanding of the importance of
geometric phases in other fields. In particular, an optical phenomenon
that had been discovered earlier by Pancharatnam  was realized to be a
manifestation of a geometric phase. The Pancharatnam-Berry phase is the
phase change that a monochromatic beam of light gains in a cyclic change
in polarization. If one maps out the changes in polarization on the
Poincaré sphere, this phase change is equal to half of the solid
angle that the polarization trajectory traces out on the sphere.
We hope to observe and measure the Pancharatnam-Berry phase with an
apparatus inspired by van Dijk et al. [3,4]. We will start with vertical
linearly-polarized light (Point A on the equator of Poincaré
sphere) and pass this through a quarter-wave plate to create circularly-
polarized light (Point B at the north or south pole). A linear polarizer
at some angle α to the vertical will then move this polarization
state to another point C on the equator. Finally, a second linear
polarizer will restore the light to its original state, point A. The
experiment consists of comparing the relative phase of the initial and
final polarization states in a Mach-Zehnder interferometer. We should
observe that the fringe shift as a function of α varies in
proportion to the solid angle mapped out on the Poincaré sphere.
We found a number of unspecified quarter-wave plates in the lab and
tested them to see which ones were effective at 632.8 nm wavelength. With
careful adjustments, the best of these produces good-quality, circularly
polarized light from the beam of our linearly polarized HeNe laser. The
interferometer was set up and fringes with good contrast were
obtained; the fringes
are susceptible to air currents, mechanical vibrations, etc, but with
care are sufficiently stable when the room is quiet. Data taking will
consist of recording the shift in fringe position
while changing the first linear polarizer from α = 0 to other values
α < 180°. The Pancharatnam-Berry phase is largest (π/2, or
1/4 of a fringe) for α = 90°, which corresponds to moving
half-way around the equator of the Poincaré sphere.
Unfortunately, as one approaches this angle the transmission of light in
the active arm of the interferometer drops to zero. Our initial
observations are a series of images of the fringe pattern with α
cycled between 0, 35 and 70 degrees. These images clearly show a fringe
shift of the expected magnitude. We are currently working to verify that
this shift is indeed due to the geometric phase.
We thank T. van Dijk and Prof. T. D. Visser for their helpful responses to
our emails. This research was supported by a grant from the National
Science Foundation (Phy-0851594).
 M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
 S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).
 T. van Dijk, H.F. Schouten, W. Ubachs and T.D. Visser, Proc. SPIE
 T. van Dijk, personal communication.